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Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square...
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Transcript of Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square...
Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric expressions involving square root.
• MM1A2b. Perform operations with square roots.
• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.
bxy
Radical Functions
• Essential questions:1. What is a radical function?2. What does the graph look like and how does
it move?3. How are they used in real life applications?
Real Life Applications
• Pythagorean Theorem• Distance Formula• Solving any equation that includes a
variable with an exponent, such as:
3
2
3
4rV
rA
Radical Expressions
• Index Radical Sign
Radicand
3 42 x
General Radical Equation
khxbay )(
Vertical stretch or compressionby a factor of |a|; for a < 0, the graph is a reflections across the x-axis
Vertical translation k unitsup for k > 0 and |k| unitsdown for k < 0
Horizontal stretch or compression by a factor of |1/b|; for b < 0, the graph is a reflection across the y-axis(b = 1 or -1 for this course)
Horizontal translation h units to the right for h > 0 and |h| units to the left if h < 0.(h = 0 for this course)
Radical Functions
• Make a (some) table(s), graph the followingfunctions and describe the transformations forx = 0, 1, 4, 9, 16 & 25.
• What transformationRules do you see fromYour graphs? )32()(
32)(
2)(
)(
xxi
xxh
xxg
xxf
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 0
1 1
4 2
9 3
16 4
25 5
0
1
4
9
16
25
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
xxf )(
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 0
1 2
4 4
9 6
16 8
25 10
02
12
42
92
162
252
xxf 2)(
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 -3
1 -1
4 1
9 3
16 5
25 7
302
312
342
392
3162
3252
32)( xxf
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 3
1 1
4 -1
9 -3
16 -5
25 -7
)32(1)( xxf
02
12
42
92
162
252
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
Radical Functions
• State an equation that would make the square root function shrink vertically by a factor of ½ and translate up 4 units.
• How would we reflect the above equation across the y-axis?
• Make the “x” negative
45.0 xy
Domain & Range: Radical Functions
• State the domain,range, and intervalsof increasing anddecreasing for each function.
)32()(
32)(
2)(
)(
xxi
xxh
xxg
xxf
Graphing Radical Functions Summary
• Transformations for radical functions are the same as polynomial functions.
• The domain of the parent function is limited to {x | x 0} (the set of all x such that x 0)
• The range of the parent function is limited to {y | y 0} (the set of all y such that y 0)
• The domain and range may change as a result of transformations.
• The parent radical function continuously increases from the origin.
Simplifying Radical Expressions• Square and square root
are inverse functions, butthe square root has to bepositive
oddisnifxx
evenisnifxx
therefore
xifxx
xifxx
n
n
____
____
0_,
0_,
2
2
525 5)5( 2 552
636 6)6( 2 662
Simplifying Radical Expressions
baab * abba *or
• “Simplify” a radical means to:1. Take all the perfect squares out of the
radicand.Simplify:
32 150 24*6 3*6
Simplifying Radical Expressions
• “Simplify” a radical means to:2. Combine terms with like radicands
• Must have the same radicand to be able to add or subtract radials
• Simplify:
8322 )3323(322
Simplifying Radical Expressions Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
• “Simplify” a radical means to:3. Do not leave a radical in the denominator
• Simplify:
2
6
8
2
b
a
b
a
10
27
2
Simplify Expressions via Conjugates
•Remember: (a+ b) is the conjugate of (a – b)•We get conjugates when we factor a perfect square minus a perfect square.•We also get conjugates other times. Simplify:
)75)(75( )23)(23(
Simplify Expressions via Conjugates
Simplify:
85
3
55
53
Summary Important Operations • Square and square root
are inverse functions, butthe square root has to bepositive
• Product Property of Square Roots: If a ≥ 0 and b ≥ 0: • Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
• b
a
b
a
baab *
0_,
0_,
2
2
xifxx
xifxx
Summary Simplification Rules• To “simplify” a radical means to:1. Take all the perfect squares out of the
radicand.2. Combine terms with like radicands3. Do not leave a radical in the denominator
Simplifying Radical Expressions
• Homework page 144, # 3 – 24 by 3’s and 25 & 26
Warm-upSimplify1.
2.
Solve.
3.
4.
4 7 125 80
3 3 2 3( )
x 2 0
4 7 5
3 6 3
4
3 2 3 7x 6
Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric expressions involving square root.
• MM1A2b. Perform operations with square roots.
• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.
bxy
Radical Functions
• Today’s essential questions:1. How do we find the solution of a radical
function?2. How are they used in real life applications?
12.3 Solving Radical Equations 1.Get the radical on one side.
2. Square both sides of the equals sign.
3. Solve for the variable.
4. Check your answer. IF the answer doesn’t check, then “no solution.”
EXAMPLE 3:
10 6 2 x100 6 2 x
10 6 17 2 ?
( ( ) )
102 6 x
x 17
10 10
EXAMPLE 1:
3 21 0x
3 21x 3 49 21 0
?
3 7 21 0( )?
42 0x 7x 49
21 21 0 ?
Your Turn – Solve with your
neighbor:
3 2 14x 3 12x
3 16 2 14 ?
3 4 2 14( )?
x 4x 16
12 2 14 ?
14 14
• The process is the same:1. Get a radical on one side.2. Square both sides of the equal sign.3. Solve for the variable.4. Repeat as necessary5. Check your answer. IF the answer does
not check, then there is NO SOLUTION for that answer
What if > One Radical?
Example 3:
19357 xx
19357 xx
244 x
6x
19)6(35)6(7
3737
1918542
?
?
Example 3:
242 xx
2422 xx
02422 xx
4
_6
x
orx
24)6(26
66
24126
?
?
046 xx24)4(24
?
?2484
164
Your Turn – Solve
Individually:
3212 xx
96212 2 xxx
01242 xx
2
_6
x
orx
3621)6(2
39 32112
?
?
026 xx ?
?
55
3221)2(2
5214
Practice with Tic-Tac-Toe• The object is to get three in a row.• Work together in designated pairs.• Notice: Different problems have
different points.• Your score will be the three in a row
you solve with the most points
Practice• Page 148, # 3 – 30 by 3’s and 31 & 32